a heuristic for two machine open shop specially structured scheduling

International Journal of Pure and Applied Mathematics Volume 86 No. 5 2013, 767-778 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v86i5.1

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A HEURISTIC FOR TWO MACHINE OPEN SHOP SPECIALLY STRUCTURED SCHEDULING PROBLEM TO MINIMIZE THE RENTAL COST, INCLUDING JOB BLOCK CRITERIA Deepak Gupta1 , Shashi Bala2 , Payal Singla3 § 1,2,3 Department

of Mathematics Maharishi Markandeshwar University Mullana, Ambala, 133001, Haryana, INDIA Abstract: The present paper considers a more practical problem of scheduling n jobs in a two machine specially structured open shop to minimize the rental cost. Further the processing time of jobs are associated with their respective probabilities including job block criteria. In most of literature the processing times are always considered to be random, but there are significant situations in which processing times are not merely random but bear a well defined structural relationship to one another. The objective of this paper is to minimize the rental cost of machines under a specified rental policy. A numerical illustration is given to support the algorithm. AMS Subject Classification: 90B35, 90B30 Key Words: open shop scheduling, rental policy, processing time, utilization time, makespan, idle time 1. Introduction In a scheduling problems for shop processing system, each job has to be processed on a specific machine. A job passing through the machines following a certain order is known as the processing route. If the processing routes are not given in advance, but have to be chosen, the processing system is called the open shop. If the processing routes are fixed beforehand and are same for all Received:

May 16, 2012

§ Correspondence

author

c 2013 Academic Publications, Ltd.

url: www.acadpubl.eu

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the jobs, the system is called flow shop. Such an open shop environment arise in many industrial application viz, automobile repairs, quality control centres, semiconductor manufacturing, setellite communication etc. Johnson [1] gave a procedure to obtain the optimal sequence for n-jobs, two three machines flow shop scheduling problem with an objective to minimize the makespan. the work was developed by Gupta, J.N.D. [4],Maggu and Dass [9], Gupta Deepak [8], Anup [18], V.A. Strusevich [14,16], Singh T.P. [12], Yoshida and Hitomi [10] etc. by considering various parameter. Singh T.P and Gupta Deepak [5] made an attempt to study the optimal two stage open shop scheduling in which processing time is associated with their respective probabilities including job block criteria. Gupta Deepak, Sharma and Shashi [20] introduce the concept of specially structured flow shop scheduling to minimize the rental cost of machines. In which processing times are associated with probabilities. The present paper is an attempt to minimize the rental cost of machines by introducing the concept of job restriction under specially structured open shop scheduling. The idea of job restrictions has a practical significance to create a balance between the cost of providing priority in service to customers and cost of giving service with non priority customers thereby the problem becomes wider and more applicable in a production concern. 2. Practical Situation Many applied and experimental situation occur in our day to day working in factories and industrial concern when we have to restrict the processing of some jobs. The practical situation may be taken in a production industry; manufacturing industry etc. where some jobs has to give priority over others. In the era of globalization or global uncertainties, to meet the challenges of the business, one does not always have enough funds to invest in advanced machines to update the technology. Under such circumstances the machines has to be taken on rent. Rental of machines is an affordable and quick solution for having the equipment and up gradation to new technology. 3. Notations Sk Mj aij pij

: Sequence obtained by applying Johnsons procedure, k = 1, 2, 3.... : Machine j= 1,2 : Processing time of ith job on machine Mj : Probability associated to the Processing time aij

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Aij : Expected Processing time tij (Sk ) : transportation time of ith job to transpose from one machine to another machine Ui (Sk ) : Utilization time for which machine Mj is required R(Sk ) : Total rental cost for the sequence(Sk ) of all machine CT (Sk ): Total completion time of the jobs for sequence (Sk ) Ci : Rental cost of ith machine β : Equivalent job block.

4. Definition Completion time of ith job on machine Mj is denoted by tij and is defined as: tij = max (ti−1,j , ti,j−1 )+aij xpij for j≥2 = max (ti−1,j , ti,j−1 )+ Aij , where Aij = Expected processing time of ith job on j th machine.

5. Rental Policy(P) The machines will be taken on rent as and when they are required and are returned as and when they are no longer required. i.e. the first machine will be taken on rent in the starting of processing the jobs, second machine will be taken on rent at time when first job is completed on the first machine.

6. Problem Formulation Let n jobs 1,2,..,n be processed on machines two machines M1 and M2 in any order i.e. the jobs will be processed first on M1 and then on M2 or first on M2 and then on M1 under the specified rental policy. Let aij be the processing time of ith job on (i = 1,2,..,n) on machine Mj and pij be the probabilities associated with aij . Aij be the expected processing time of ith job on machine Mj (j = 1,2,n) such that either Ai1 ≥ Ai2 or Ai1 ≤ Ai2 . Our aim is to find the optimal or near optimal sequence Sk of the jobs which minimize the rental cost of machines.The mathematical model of the given problem in matrix form can be stated as:

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D. Gupta, S. Bala, P. Singla jobs i 1 2 3 n

Machine ai1 a11 a21 a31 an1

A pi1 p11 p21 p31 pn1

Machnie ai2 a12 a22 a32 an2

B pi2 p12 p22 p32 pn2

Table 1

P Mathematically, the problem is stated as: Minimize R(Sk ) = Ai1 × C1 + U2 (Sk ) × C2 . Subject to constraint: Rental Policy(P). Our objective is to minimize rental cost of machines while minimizing the utilization time.

7. Assumptions 1. Two jobs cannot be processed on a single machine at a time. 2. Jobs are independent to each other. 3. Per-emption is not allowed i.e. once a job started on a machine, the process on that machine cannot be stopped unless the job is completed. 4. Either the processing time of the ith job of machine M1 is longer than the expected processing time of j th job on machine M2 or the processing time ith job on machine M1 is shorter than the expected processing time of j th job on machine M2 for P P all i,j. either Ai1 ≥ Ai2 or Ai1 ≤ Ai2 5. pi1 =1, pi2 =1 for all i. 6. Let n jobs be processed through two machines M1 and M2 in order M1 M2 and in order M2 M1 . 7. Machine break down is not considered.

8. Theorem If for all Ai1 ≤ Aj2 for all i 6= j, then k1 , k2 ...kn is a monotonically decreasing P i, j, P sequence, where Kn = Ai1 - Ai2 . Proof. P Let Ai1 P≤ Aj2 for all i, j, i 6= j i.e.,max Ai1 ≤ minAj2 for all i, j, i 6= j Let Kn = Ai1 - Ai2 . Therefore, we have k1 =A11 .

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Also K2 = A11 + A21 − A12 = A11 + (A21 − A12 ) ≤ A11 (as A21 ≤ A12 ), so K1 ≤K2 . Now, K3 = A11 + A21 + A31 − A12 − A22 = A11 + A21 − A12 + (A31 − A22 ) = K2 + (A31 − A22 ) ≤ K2 (as A31 ≤ A22 ). Therefore, K3 ≤ K2 ≤ K1 or K1 ≥ K2 ≥ K3 . Continuing in this way, we can haveK1 ≥ K2 ≥ K3 ......≥ Kn , a monotonically decreasing sequence. Corollary 1. The total rental cost of machines is same for all the sequences if Ai1 ≤ Aj2 for all i, j, i 6= j. P P Proof. The total elapsed time T (S) = Ai2 + K1 = Ai2 +A11 . It implies that under rental policy P the total elapsed time on machine M2 is same for all the sequences thereby the rental cost of machines is same for all the sequences. 9. Theorem If Ai1 ≥ Aj2 for all i,P j, i 6=Pj, then k1 , k2 ...kn is a monotonically increasing sequence, where Kn = Ai1 - Ai2 P P Proof. Let Kn = Ai1 - Ai2 . Let Ai1 ≥ Aj2 for all i, j, i 6= j, i.e. min Ai1 ≥ maxAj2 for all i, j, i 6= j. Here k1 =A11 , K2 = A11 + A21 -A12 = A11 + (A21 - A12 ) ≥ k1 ( as A21 ≥ A12 ) so K2 ≥K1 Now, K3 = A11 + A21 + A31 -A12 - A22 = A11 + A21 -A12 +( A31 - A22 ) = K2 +( A31 -A22 ) ≥ K2 ( as A31 ≥ A22 ) Hence, K3 ≥ K2 ≥ K1 . Continuing in this way, we can haveK1 ≤ K2 ≤ K3 ......≤ Kn , a monotonically increasing sequence. Corollary 2. The total elapsed time of machines is same for all the possible sequences. Proof. The total elapsed time

772 T (S) =

D. Gupta, S. Bala, P. Singla X

Ai2 + Kn =

X

X X Ai2 + ( Ai1 − Ai2 ) X X X X = Ai1 + ( Ai2 − Ai2 ) = Ai1 + An2 .

Therefore total elapsed time of machines is same for all the sequences if Ai1 ≥ Aj2 for all i, j, i 6= j. 10. Algorithm Step 1: Calculate the expected processing time Aij = aij × pij ∀i,j. Step 2: Take the equivalent job (k, m) and calculate the processing time Gβ and Hβ on guide lines of Maggu and Dass (1982) as follows: Gβ = Gk + Gm - min(Gm , Hk ) and Hβ = Hk + Hm - min(Gm , Hk ) Step 3: Define a new reduced problem with processing times Gi and Hi as defined in step 1 and jobs (k, m) are replaced by single equivalent job β with processing time Gβ and Hβ as defined in step 2. Step 4:Check the condition either Gi ≥ Hj or Gi ≤ Hj , for all i and j, i 6= j if conditions hold good, go to step5, else the results will coincides with [5]. Step 5: Obtain the job J1 (say) having maximum processing time an first machine and job Jn (say) having minimum processing time an second machine. Step 6: If J1 6= Jn then put J1 on the first position and Jn as the last position and go to step 9, Otherwise go to step 7. Step 7: Take the difference of processing time of job J1 on first machine from job J2 (say) having next maximum processing time on first machine. Call this difference as G1 . Also, take the difference of processing time of job Jn on second machine from job Jn−1 (say) having next minimum processing time on second machine. Call the difference as G2 . Step 8: If G1 6= G2 put Jn on the last position and J2 on the first position otherwise put J1 on first position and Jn−1 on the last position. Step 9: Arrange the remaining (n-2) jobs between first job and last job in any order, thereby we get the sequences S1 ,S2 ,S3 ,...,Sr (when the order of machines is M1 →M2 and the sequences S1′ ,S2′ ,S3′ ,...,Sr′ (when the order of machines is M2 →M1 ). Step 10: Compute the total completion time CT (Sk ) and CT (Sk′ ) by computing in out table for sequences Sk and Sk′ (k= 1, 2,...,r.).

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Step 11: Calculate utilization time CT (Sk ), U2 (Sk ) and U1 (Sk′ ) of second machine as follows: U2 (Sk ) = CT (Sk ) - A11 (Sk ) ; k=1,2,. r. and U1 (Sk′ ) = CT (Sk′ ) -A12 (Sk′ ) ; k=1,2,. r. P Step 12: Find rental cost R(S1 )= Ai1 (S1 ) x C1 + U2 (S1 ) x C2 , where C1 and C2 are the rental cost per unit time of machine M1 and M2 respectively. P Step 13: Find rental cost R(S1′ )= Ai2 (S1′ ) x C2 + U2′ (S1′ ) x C1 where C1 and C2 are the rental cost per unit time of machine M1 and M2 respectively. Step 14: Find R(S)=min (R(S1 ), R(S1′ )).

11. Numerical Illustration Consider 6 jobs, 2 machines problem to minimize the rental cost, processing time associated with their respectively probabilities as given in following table and job 2,5 are to be processed as a group job (2,5). The rental cost per unit time for machines M1 and M2 are 10 and 5 units respectively. Our objective is to obtain optimal schedule and order of machines to minimize the total production time subject to minimization of the rental cost of the machines, under the rental policy p. jobs Machine M1 Machnie M2 i ai1 pi1 ai2 pi2 1 40 0.2 75 0.2 2 30 0.3 80 0.2 3 45 0.1 85 0.2 4 35 0.2 90 0.2 5 25 0.1 95 0.1 6 50 0.1 100 0.1 Table 2 Solution. As per Step 1: The expected processing time for machines M1 and M2 is as follow

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D. Gupta, S. Bala, P. Singla job 1 2 3 4 5 6

Ai1 8.0 9.0 4.5 7.0 2.5 5.0

Ai2 15.0 16.0 17.0 18.0 9.5 10.0

Table 3 when the order of machines is M1 →M2 . As per Step 2: Here β=(2,5), Gβ = G2 + G5 - min(G5 , H2 )= 9.0 and Hβ = H2 + H5 - min(G5 , H2 ) = 23.0 job 1 β 3 4 6

Gi 8.0 9.0 4.5 7.0 5.0

Hi 15.0 23.0 17.0 18.0 10.0

Table 4 As per step 4 Gi ≤ Hi for all i, j, i6=j. Also max Gi = 9.0 which is for job β i.e. J1 = β And min Hi = 10.0 which is for job 3 i.e. Jn = 6 Since J1 6= Jn . As per Step 6, job β will be on first position and job 6 will be on the last position. Therefore as per step 9 the possible optimal sequences are S1 : β -1 - 3 - 4 - 6, S2 : β - 3 - 1 - 4 -6, S3 : β - 1 - 4 - 3 - 6, S4 : β - 3 - 4 - 1 - 6, S5 : β - 4 - 1 -3 - 6, S6 : β - 4 - 3 - 1 - 6 job i 2 5 1 3 4 6

MachineM1 In-Out 0 - 9.0 9.0 - 11.5 11.5 - 19.5 19.5 - 24.0 24.0 - 31.0 31.0 - 36.0

MachineM2 In-Out 9.0 - 25.0 25.0 - 34.5 34.5 - 49.5 49.5 - 66.5 66.5 - 84.5 84.5 - 94.5

Table 5 The total elapsed time is same for all these 6 possible sequence S1 , S2 ,

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S3 ,S4 , S5 , S6 The in- -out table for any of these sequences S1 , S2 , S3 ,S4 , S5 , S6 Say for S1 : 2 5 1 3 4 6 Therefore the total elapsed time P = CT (S1 )= 94.5 units and utilization time for M2 = U2 (S1 ) = 85.5 units Also Ai1 =36.0 and rental cost R(S1 ) = 36.0 10 + 85.5 5 = 787.5 units, when the order of machines is M2 →M1 . As per Step 2: Here β=(2,5) Gβ = G2 + G5 - min(G5 , H2 )= 16.5 and Hβ = H2 + H5 - min(G5 , H2 ) = 2.5. As per Step 3: New reduced problem is as follows job 1 β 3 4 6

Gi 15.0 16.5 17.0 18.0 10.0

Hi 8.0 2.5 4.0 7.0 5.0

Table 6 As per step 4 Gi and Hj for all i, j. Also max Gi = 18.0 which is for job 4 and min Hi = 2.5 which is for job β i.e. Jn = β Since J1 6= Jn . As per step 6, job 4 will be on first position and job β will be on the last position. As per step 9 the possible optimal sequences are S1′ : 4-1-5-3-β, S2′ : 4-1-35-β,..., S6′ :4-3-6-1- β The total elapsed time is same for all these 6 possible sequence S1′ , S2′ ,...,S6′ The in- -out table for any of these sequences S1′ , S2′ ,...,S6′ Say for S1′ :4-3-1-6-2-5. job i 4 3 1 6 2 5

MachineM2 In-Out 0 - 18 18 - 35 35 - 50 50 - 60 60 - 76 76 - 85.5

MachineM1 In-Out 18 - 25.0 35 - 39.5 50 - 58 60 - 65 76 - 85 85.5 - 88.0

Table 7 Therefore the total elapsed timeP= CT (S1′ )= 88.0 units and utilization time Ai2 =85.5 and rental cost R(S1′ ) = 1127.5 for M1 = U1 (S1′ ) = 70.0 units Also ′ units R(S)=min (R(S1 ), R(S1 ))=787.5 units=R(S1 ).

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We conclude that optimal schedule S1′ : 4 -3 - 1 - 6 -2 - 5, which is for order Machine M1 to MachineM2 gives minimum rental cost. 12. Remarks 1. If the assumption 4 is not considered the result tally with [5]. 2. Equivalent job formulation is associative and non commutative in nature. 3. The study on n*2 open shop scheduling may be further extended by including various parameters such as weightage of jobs, transportation etc. 4. The study may further be extended for n job 3 machine open shop problem. 5. If we solve the same problem by Johnson[1] method we get optimal sequence as S=3-6-4-1-β = 3-6-4-1-2-5 for M1 to MachineM2 This gives rental cost R(S) = 837.5 Units And for order M2 to MachineM1 we get optimal sequence as S=1-4-6-3-β =1-4-6-3-2-5. which gives R(S) =1157.5. Thus rental cost is minimum forM1 to MachineM2 .

13. Conclusion The algorithm proposed in this paper to minimize the rental cost of machines gives an optimal sequence having minimum rental cost of machines. The algorithm proposed by Johnson [1] to find an optimal sequence to minimize the makespan / total elapsed time is not always corresponds to minimum rental cost the machines. Hence proposed algorithm is more efficient to minimize the rental cost of machines under a specified rental policy. We conclude that ours algorithm gives better result.

References [1] Johnson, S.M. , Optimal two and three stage Production Schedule with set-up times included. Nav Res Log.Quart., Vol. 1, 1954, pp.61–64. [2] Ignall, E., and Schrage, L. , Application of the branch and bound technique to some flow shop scheduling problems . Operation Research, Vol. 13, 1965, pp.400–412. [3] Bagga, P.C., Sequencing in a rental situation, Journal of Canadian Operation Research Society, Vol.7, 1969, pp.152–153.

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[4] Gupta, J.N.D, Optimal Schedule for specially structured flow shop, Naval Res. Logist. Quart., Vol.22,No.2, 1975, pp.255–269. [5] Gupta D.and Singh, T.P., On job block open shop scheduling, the processing time associated with probability.,J. Indian Soc. Stat. Opers. Res., Vol.XXVI No.1-4,2005,pp.91-96. [6] Harbans, Lal and Maggu, P.L. , On job block open shop scheduling problem, PAMS, Vol.XXIX, 1989, pp.45–51. [7] Gonzalez, T. and Sahni, S. , Open shop scheduling to minimize nish time, J. Assoc. Comput., March. 23, 1976, pp.665–679. [8] Singh, T. P. and Gupta D. , Minimizing rental cost in two stage flow shop, the processing time associated with probabilities including job block, Reflections de ERA, Vol. 1, No.2, 2005, pp.107–120. [9] Maggu, P.L.and Das G. , Equivalent jobs for job block in job scheduling, Opsearch, Vol.14, No.4,1977, pp.277–281. [10] Yoshida and Hitomi ,Optimal two stage production scheduling with set up times separated, AIIE Transactions,Vol. 11,No.3, 1979, pp.261-269. [11] Gupta, J.N.D, Two stage hybrid flow shop scheduling problem, J. Ope. Res. Soc., Vol.39,No.4, 1988, pp.359–364. [12] Singh, T. P. , On n2 shop problem involving job block. Transportation times and Break-down Machine times, PAMS, Vol. XXI, No.2, 2005, pp.1. [13] V.J. Rayward-Smith,and D. Rebaine, Open-shop scheduling with delays, Theoret. Inform. Appl.,1992, 439-448. [14] V.A. Strusevich, Two-machine open-shop scheduling problem with setup, processing and removal times separated, Comput. Oper. Res.,1993, 597611. [15] Amico, M.,Dell, Shop problems with two machines and time lags, Oper. Res.,1996,Vol.44, 777-787.. [16] V.A. Strusevich, An open-shop scheduling problem with a non-bottleneck machine, Oper. Res. Lett.,1997,Vol.21, 11-18. [17] Rebaine,D.,and Strusevich,V.A., Two-machine open shop scheduling with special transportation times, CASSM Rand D,1998,Vol.15, University of Greenwich, London, UK.

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[18] Anup, On two machine flow shop problem in which processing time assumes probabilities and there exists equivalent for an ordered job block, JISSOR,2005, Vol. XXIII,No.1-4. [19] T.P.Singh, Rajender kumar and Gupta Deepak, Optimal three stage production schedule the processing time and set times associated with probabilities including job block criteria, Acta Ciencia Indica, Proceedings of National Conference FACM, 2005. [20] Gupta D.,Sharma,S.,and Shashi B., Optimal three stage production schedule the processing time and set times associated with probabilities including job block criteria, Specially Structured Two Stage Flow Shop Scheduling To Minimize the Rental Cost, International Journal of Emerging trends in Engineering and Development, Vol. 1, No. 2, 2005, pp.206-215.